The news reports don’t actually state how much energy the storage device can store.

A typical phone battery has 1500 mAh of energy stored. If you charged this in 30 seconds, that would require 670 Watts. Something would probably melt in the phone.

The article mentions the student using a supercapacitor instead of just a plain boring capacitor. According to Wikipedia, it seems that the term “supercapacitor” could be used for several different devices. Some of which aren’t really capacitors. But I think this is still a great opportunity to talk about capacitors.

What is a capacitor?

In just about every introductory physics course, you are going to look at capacitors. The most basic capacitor to start with is a parallel plate air filled capacitor. It’s just two conducting plates that are very close to each other. Positive charge is on one plate and the same amount of negative charge is on the other plate.

But why? Who cares about capacitors? In the introductory physics course, capacitors are typically introduced for one main reason: the electric field inside a capacitor is essentially constant. Constant is nice. It allows you do set up situations involving electric charges with constant forces (due to the constant electric field) and simple changes in electric potential.

Of course, there are other types of capacitors than your plain old parallel plate with air in the middle. But you get the idea.

What is a capacitor used for?

Since there is an electric field inside the capacitor, there is also energy stored in the capacitor (you can use the energy density of the electric field). So obviously, a capacitor can be used to store energy. However, there are other very nice uses. Capacitors turn out to be extremely important in applications that involve changing currents – like AC-DC converters or any type of radio. If you connect a plain DC battery to a capacitor, charge will start to build up on the plates. When this happens, there is also an electric potential difference across the plates. This means that there will be less current coming out of the battery and less charge going to the plates. Technically, the capacitor will take forever to become fully charged. Maybe I can show this with a nice graph. Here is the charge on a capacitor as a function of time after being hooked to a DC battery.

Hope that helps. But what happens if you have a changing current? In this case the charge goes on and off the capacitor and it behaves differently. The higher the frequency that the current changes, the lower the electric potential difference across the capacitor. So, at high frequencies, it’s almost like it isn’t even in the circuit. Using this along with an inductor (that has high electric potential differences at high frequencies) can create circuits that act different for different frequencies.

So, you will find these capacitors in lots of different applications. They aren’t just used for energy.

How much energy can a capacitor store?

There are a few ways to calculate the energy stored in a capacitor. You could determine the electric field and use the energy density of the field. Or if you prefer, you could look at the current going into the capacitor along with the potential difference across the capacitor as it charges. Either way, the energy depends on the physical dimensions of the capacitor and the charge on the capacitor. You can derive the energy for homework, but here is the result.

The more charge, the greater the energy. You can also increase the energy stored by increasing the capacitance of the capacitor or by increasing the voltage across the capacitor. Be careful though. If you increase the voltage too much, there will be a large charge on the plates. One of two bad things can happen. The first is that the stuff in the middle of the capacitor could become a conductor. For air, this happens at an electric field strength around 3 x 106 V/m. The other thing is an implosion. The two plates are attracted to each other since they have opposite charges. At some point they will just move towards each other (if there is stuff in between, it gets squished out).

What is the k in the expression above? This is the dielectric constant and is for the case were stuff is in the middle of the plates.

What would a capacitor for your phone look like?

Suppose you wanted to replace your phone battery with a capacitor. This would be a difficult thing to do since as you use energy from the capacitor, it would decrease in potential. Batteries do this too, but not so much that it matters. Let’s proceed anyway with our design process. First some assumptions.

I want the capacitor to be sort of equivalent to a 1500 mAh battery at 4 volts (the iPhone 5 battery is 1400mAh at 3.8 Volts). That would be an energy of 2.16 x 104 Joules.

I want this capacitor to have a maximum voltage of 6 volts. I just made that up.

With these parameters, I can calculate the value of the capacitance for this energy storage.

That’s a pretty huge value for a capacitor. A typical capacitance will commonly be in the microFarads and not 1000 Farads. However, let’s proceed. How big would this parallel plate capacitor be? First, I want the two plates to be as close together as possible. One limit to this spacing will be the breakdown voltage of the dielectric material. According to Wikipedia, mica can have a breakdown voltage as high as 300 x 106 V/m.

At a potential difference of 6 volts, the plates can be 2 x 10-8 meters apart. That seems crazy close – but oh well. I am going with that number anyway. Now, I just need to solve for the area of my capacitor.

If this is a square capacitor, that would have dimensions of 672 meters by 672 meters. Try putting that in your phone. Sure you could roll it up, but I still think it would be HUGE.

How long would this capacitor take to charge up? Let’s say that the resistance of the connecting wires is 10 Ohms (just a guess) and that I apply a charging potential of 20 Volts. For a charging capacitor, I can write (without deriving it):

Solving for t, I get 4952 seconds (1.4 hours). See – I used a trick here. I want the capacitor at 6 volts, so I applied a 20 volt battery to charge it. Oh, I also cheated and assumed an uncharged capacitor (which it wouldn’t be). However, it would still take some time to charge.

How long would this capacitor-energy storage last? Let’s say that you can use this until it’s down to a potential of 2.5 volts. If it starts at 6 volts, how long would this take? Let’s say that I have a phone that uses an average of 0.5 watts. In order to keep this power, I am going to have to change the resistance in the phone as the voltage across the battery decreases. You know what this means, right? I am going to have a make a numerical model to calculate this time.

Here is my plan. Of course I will break the whole process into short time intervals. During each interval, I will:

Determine the charge on the capacitor and thus the potential difference across the capacitor.

Determine the resistance needed to produce a power of 0.5 watts.

Calculate the current with this potential difference and resistance.

Use this current to determine the change in charge on the capacitors. BOOM.

This lasts for 35,720 seconds or 9.9 hours. Not too bad. However, remember that this is for a HUGE capacitor. Also, I am making two assumptions. First that the phone can operate over a wide range of voltages. Second, that the phone can change something such that it gets a constant power (not likely).

Back to the science project

I want to be clear about something. I think this student’s project is probably pretty cool (I don’t know enough of the details to say for sure). However, I have doubts that you could really charge a phone in 30 seconds in any useful way. It clearly wouldn’t work out too well with a plain capacitor.